Problem: A stone fell from the top of a cliff into the ocean. In the air, it had an average speed of $16$ $\text{m/s}$. In the water, it had an average speed of $3$ $\text{m/s}$ before hitting the seabed. The total distance from the top of the cliff to the seabed is $127$ meters, and the stone's entire fall took $12$ seconds. How long did the stone fall in the air and how long did it fall in the water? The stone fell in the air for
Explanation: Let $x$ represent the time (in seconds) that the stone fell in the air and let $y$ represent the time (in seconds) that it fell in the water. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that the stone fell at an average speed of $\textit{16 m/s}$ in air, at an average speed of $\textit{3 m/s}$ in water, and that it fell a total distance of $\textit{127}$ meters. How can we model this sentence algebraically? The total distance the stone fell in air can be modeled by $16x$, and the total distance it fell in water can be modeled by $3y$. Since these together add up to $127$, we get the following equation: $16x+3y=127$ We are also given that the stone's entire fall took $\textit{12}$ seconds. This can be expressed as: $x + y =12$ Now that we have a system of two equations, we can go ahead and solve it! We can now solve the system of equations by the elimination method. Note that the coefficient of $y$ in the first equation, $3$, is exactly $3$ times the coefficient of $y$ in the second equation, $1$. Therefore, we can multiply the second equation by ${-3}$ in order to eliminate $y$. $ \begin{aligned} {-3}\cdot x+({-3})\cdot y&={-3}\cdot12\\\\ -3x-3y&=-36\end{aligned}$ Now we can eliminate $y$ : $\begin{aligned}16x+{3y}&=127\\\\ {+}\ -3x-{3y}&=-36\\ \hline\\ 13x+0 &=91 \end{aligned}$ When we solve the resulting equation, we find that $x =7$, which we can substitute into $x+y=12$ to find that $y=5$. Recall that $x$ denotes the time the stone fell in air and $y$ denotes the time it fell in water. Therefore, the stone fell in the air for $\textit{7}$ seconds and fell in the water for $\textit{5}$ seconds.